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Subgame Perfect Implementation with Almost Perfect Information and the Hold-Up Problem

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Subgame Perfect Implementation with Almost Perfect Information and the Hold-Up Problem NBER WORKING PAPER SERIES SUBGAME PERFECT IMPLEMENTATION WITH ALMOST PERFECT INFORMATION AND THE HOLD-UP PROBLEM Philippe Aghion Drew Fudenberg Richard T. Holden Working Paper 15167 http://www.nber.org/papers/w15167 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 July 2009 We owe special thanks to Oliver Hart for numerous discussions, and to Johannes Horner and Andy Skrzypacz for pointing out an error in a previous version. We are also grateful to Mathias Dewatripont, Bob Gibbons, Philippe Jehiel, John Moore, Roger Myerson, Andrew Postlewaite, Olivier Tercieux, Jean Tirole, Ivan Werning and Muhamet Yildiz for helpful discussions and comments. Michael Powell provided excellent research assistance. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer- reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2009 by Philippe Aghion, Drew Fudenberg, and Richard T. Holden. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Subgame Perfect Implementation with Almost Perfect Information and the Hold-Up Problem Philippe Aghion, Drew Fudenberg, and Richard T. Holden NBER Working Paper No. 15167 July 2009 JEL No. C72,C73,D23,L22 ABSTRACT The foundations of incomplete contracts have been questioned using or extending the subgame perfect implementation approach of Moore and Repullo (1988). We consider the robustness of subgame perfect implementation to the introduction of small amounts of asymmetric information. We show that Moore- Repullo mechanisms may not yield (even approximately) truthful revelation in pure or totally mixed strategies as the amount of asymmetric information goes to zero. Moreover, we argue that a wide class of extensive-form mechanisms are subject to this fragility. Philippe Aghion Richard T. Holden Department of Economics University of Chicago Harvard University Booth School of Business 1805 Cambridge St Room 528 Cambridge, MA 02138 5807 S Woodlawn Ave and NBER Chicago,IL 60637 [email protected] and NBER [email protected] Drew Fudenberg Department of Economics Harvard Unviersity 1805 Cambridge St Cambridge, MA 02138 [email protected] 1 Introduction Grossman and Hart (1986) argued that in contracting situations where states of nature are observable but not veri…able, asset ownership (or vertical integration) could help limit the extent to which one party can be held up by the other party, which in turn should encourage ex ante investment by the former. However, vertical integration as a solution to the hold-up problem has been questioned in various papers1 which all use or extend the subgame perfect implementation approach of Moore and Repullo (1988). In particular, Maskin and Tirole (1999a)2 argue that although parties may have di¢ culty foreseeing future physical contingencies they can write contracts which specify ex ante the possible payo¤ contingencies. Once the state of the world is realized, the parties can “…ll in”the physical details. The latter step is subject to incentive-compatibility considerations. That is, each agent must be prepared to specify the details truthfully. Maskin and Tirole achieve this through a 3-stage subgame perfect implementation mechanism which induces truth-telling by all parties as the unique equilibrium outcome3. In this paper, we consider the robustness of the Moore-Repullo (MR) mechanism to the introduction of small amounts of asymmetric information. We …nd that the MR mechanism may not yield even approximately truthful revelation in pure or totally mixed strategies as the amount of informational asymmetry goes to zero. Moreover, we show that this non- robustness result does not require informational asymmetries on agents’types (call it the introduction of crazy types): small deviations from symmetric information on the buyer’s willingness to pay for the good or on the seller’s production cost, su¢ ce to generate this non-robustness result. This, in turn, has important implications for the debate on the foundations of incomplete contracts: namely, while asymmetric information about agents’ types can eliminate the hold-up problem at the same time as it introduces undesirable equilibria in sequential mechanisms, asymmetric information about valuation and costs is 1 For example, see Aghion-Dewatripont-Rey (1999) and recently Maskin-Tirole (1999a, 1999b). 2 See also Maskin and Tirole (1999b). 3 Whereas Nash implementation (see Maskin 1977, 1999) does not guarantee uniqueness. 2 shown to preserve the hold-up problem even though it perturbs the MR mechanism. We proceed in several steps. In Section 2 we introduce a simple example of ex-post bargaining and exchange drawn from Hart and Moore (2003)4 to illustrate our …rst point on the robustness of the MR mechanism to the introduction of small amounts of asymmetric information. More precisely, we modify the signal structure of the game by assuming that each player receives private signals about the true value of the good, instead of knowing it perfectly; thus the value is “almost common knowledge”in the sense of being common p-belief (Monderer and Samet (1989)) for p near 1. Our main …nding here is that the simple subgame- perfect implementation mechanism à la MR for this example, does not yield approximately truthful revelation in either pure or totally mixed strategies as the correlation between the private signals and the true value of the good becomes increasingly perfect, although truthful revelation can be approximated by a partially mixed equilibrium. The basic idea behind this result is that even a small amount of uncertainty at the interim stage, when players have observed their signals but not yet played the game, can loom large ex post once a player has sent a message. This is closely related to the observation that backwards induction and related equilibrium re…nements are not in general robust to perturbations of the information structure (see Fudenberg, Kreps and Levine (1988), Dekel and Fudenberg (1990) and Borgers (1994)) so that the predictions generated under common knowledge need not obtain under almost common knowledge. However, in this example we restrict attention to informational asymmetries about the value of the good as opposed to the more general perturbations considered in the robustness literature. More speci…cally, in our modi…cation of the Hart-Moore-Repullo example, the Seller produces a good whose valuation is stochastic, and may be high or low. Each contracting party gets a private and yet almost perfect signal about the good’s valuation; the players have a common prior on the joint distribution of values and signals. The Moore-Repullo mechanism requests that one party, say the Buyer, make an announcement about the value of the good, and then the 4 This example itself illustrates the mechanism in Moore and Repullo (1988, Section 5). 3 Seller may either challenge or not challenge the Buyer’s announcement. Obviously, under perfect information, the Buyer’sannouncement contains no information which the Seller did not have ex ante. However, when each player receives a private signal about the value of the good, the Buyer’sannouncement does contain information about her own signal of the good’s valuation. The Seller will then update his belief about the value of the good, on the basis of both, his own signal and the announcement made by the Buyer. And the resulting Bayesian updating is what causes the subgame implementation logic to break down mechanism to break down. For example, if the two parties commit to play the MR mechanism and yet the Buyer announces a low value for the good, then the Seller will update his beliefs towards the notion that the true value of the good is indeed low. Consequently, the Seller will not challenge the Buyer by fear of being …ned under the MR mechanism for having unfairly challenged the Buyer. In Section 3 we extend our analysis to a general setting with n states of nature and trans- ferable utility, and we show that in this setting there exist natural social choice functions which cannot be implemented in any totally mixed equilibrium of perturbed MR mecha- nisms with arbitrarily small amount of private information about the value of the good. In Section 4 we move beyond Moore-Repullo mechanisms and ask whether the same logic can apply to any extensive-form mechanism. Here we make a more general but weaker claim: for any game induced by an extensive form-mechanism, there exists a nearby game with almost perfect information in which at least one equilibrium is “undesirable”, i.e. does not induce truth-telling by the contracting parties. While this conclusion can be easily established if we allow for private signals about payo¤s, based on Fudenberg, Kreps and Levine (1988), henceforth FKL5, we also prove non-robustness to the common p-belief perturbation con- sidered in Section 2 when restricting attention to pure strategies and three-stage sequential mechanisms.6. 5 FKL show that any Nash equilibrium is the limit of strict (and thus sequential) equilibria when a small change is made to prior beliefs. So if agents have any doubts about the payo¤ structure, extensive-form mechanisms cannot robustly improve on Nash implementation. 6 Parallel work by Kunimoto and Tercieux (2009), who show that only Maskin-monotonic social choice 4 Thus, Section 4 suggests that if we start from any subgame implementation mechanism under perfect information, one could show the existence of at least one undesirable equi- librium with arbitrarily small perturbations of the information structure, whereas Section 3 shows that for MR mechanisms, this perturbation leads to uniqueness of an undersir- able equilibrium.
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